How to make an inventory system for the Blender Game Engine
Okay, what you are asking is quite a big task and you first need to make a saving function so that items are saved for later on and don't just disappear. That needs python, and luckily this exists: https://www.youtube.com/watch?v=ffws4ijjk-0

Then you need to make a second scene that has the HUD, which is basically just an image of some boxes where items would go. Then you would add an empty in each slot, and then set it up so that when you pick up an item (or interact with an item) the empty would add the object into the HUD slot where the empty is located. And then you would make a property for each item you can possibly get, so that when you have the item it can add 1 to the property. When you use the item, it would take away the 1 from the property and get rid of it from the HUD, by ending the object that represents the item. Finally, you need to add a property that keeps track of how many items you have. For example, if the HUD could hold 6 items, then you need to set it up so when you try to pick up a 7th item, it would not let you because the value is 6 already. Once this value is decreased to 5 or less, then it would let you pick it up.

This is mostly just my ideas and I have't actually ever done this, but I strongly believe it would work. It might be a bit tricky setting up which empty adds the HUD object though.